Exact rate of convergence of the expected $W_{2}$ distance between the empirical and true Gaussian distribution

Philippe Berthet (1) , Jean Claude Fort (2)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5

We study the Wasserstein distance W 2 for Gaussian samples. We establish the exact rate of convergence log log n/n of the expected value of the W 2 distance between the empirical and true c.d.f.'s for the normal distribution. We also show that the rate of weak convergence is unexpectedly 1/ √ n in the case of two correlated Gaussian samples.

Mots clés: Gaussian empirical c.d.f; quadratic Wasserstein distance; Central limit theorem; Empirical processes; Strong approximation

Domaines: Probabilités [math.PR]; Statistiques [math.ST]

Philippe Berthet, Jean Claude Fort. Exact rate of convergence of the expected $W_{2}$ distance between the empirical and true Gaussian distribution. Electronic Journal of Probability, 2020, 25, pp.12. ⟨10.1214/19-EJP410⟩. ⟨hal-02495832⟩